这个词组with
me没啥好说的，关键就是suffer的理解。
这儿，suffer作为不及物动词，其意义为：（柯林斯词典）
2.V-I
If
you
suffer
from
an
illness
or
from
some
other
bad
condition,
you
are
badly
affected
by
it.
(因疾病等糟糕的处境而)
受苦
例：
He
was
eventually
diagnosed
as
suffering
from
terminal
cancer.
他最后被诊断为癌症晚期。
4.V-I
If
you
suffer,
you
are
badly
affected
by
an
event
or
situation.
受难
例：
There
are
few
who
have
not
suffered.
几乎无人免于受难。
这两个意义是相近的，都是受苦，遭罪的意思。
所以，suffer
with
me的意思就是“与我一起经历苦难”。
在结婚时牧师的那段台词里，这个词时有出现
夫妻应该同甘苦，***患难。
不过貌似现在中国的夫妻里，***患难还是不要谈为好。